To determine the zeros of the function [tex]\( f(x) = -2x^3 + 13x^2 - 22x + 8 \)[/tex], we need to find the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex].
The given polynomial is:
[tex]\[ f(x) = -2x^3 + 13x^2 - 22x + 8 \][/tex]
To find the zeros, we set the function equal to zero and solve for [tex]\( x \)[/tex].
[tex]\[ -2x^3 + 13x^2 - 22x + 8 = 0 \][/tex]
To solve this cubic equation, we can use methods such as factoring, synthetic division, or the Rational Root Theorem to identify potential rational roots. By substituting prospective rational roots into the polynomial, we can determine whether they satisfy the equation [tex]\( f(x) = 0 \)[/tex]. Here's the step-by-step process:
1. Identify potential rational roots:
The Rational Root Theorem suggests that any rational root, expressed as [tex]\( \frac{p}{q} \)[/tex], is a factor of the constant term (8) divided by a factor of the leading coefficient (-2).
Factors of [tex]\( 8 \)[/tex]: ±1, ±2, ±4, ±8
Factors of [tex]\( -2 \)[/tex]: ±1, ±2
Possible rational roots:
[tex]\[
\pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{2}{2} = \pm 1 \text{ (already included)}, \pm \frac{4}{2} = \pm 2 \text{ (already included)}, \pm \frac{8}{2} = \pm 4 \text{ (already included)}
\][/tex]
2. Test the potential rational roots:
By substituting the potential rational roots into the polynomial, we can verify which values satisfy the equation [tex]\( f(x) = 0 \)[/tex]:
- [tex]\( x = \frac{1}{2} \)[/tex]
- [tex]\( x = 2 \)[/tex]
- [tex]\( x = 4 \)[/tex]
When [tex]\( x = \frac{1}{2} \)[/tex]:
[tex]\[
f\left(\frac{1}{2}\right) = -2\left(\frac{1}{2}\right)^3 + 13\left(\frac{1}{2}\right)^2 - 22\left(\frac{1}{2}\right) + 8 = 0
\][/tex]
When [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = -2(2)^3 + 13(2)^2 - 22(2) + 8 = 0
\][/tex]
When [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = -2(4)^3 + 13(4)^2 - 22(4) + 8 = 0
\][/tex]
Since the values [tex]\( x = \frac{1}{2} \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( x = 4 \)[/tex] satisfy [tex]\( f(x) = 0 \)[/tex], they are the zeros of the function.
Therefore, the zeros of the function [tex]\( f(x) = -2x^3 + 13x^2 - 22x + 8 \)[/tex] are:
[tex]\[ x = \frac{1}{2}, \, x = 2, \, \text{and} \, x = 4 \][/tex]
